A space craft of mass $M$ is moving with velocity $V$ and suddenly explodes into two pieces. A part of it of mass m becomes at rest, then the velocity of other part will be

A bullet weighing $10 \,g$ and moving with a velocity $300 \,m / s$ strikes a $5 \,kg$ block of ice and drop dead. The ice block is kept on smooth surface. The speed of the block after the collision is ........ $cm / s$

Given in Figures are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

Two particles, $1$ and $2$ , each of mass $m$, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $x_0$, are oscillating with amplitude a and angular frequency $\omega$. Thus, their positions at time $t$ are given by $x_1(t)=\left(x_0+d\right)+a \sin \omega t$ and $x_2(t)=\left(x_0-d\right)-$ $a$ sin $\omega t$, respectively, where $d>2 a$. Particle $3$ of mass $m$ moves towards this system with speed $u_0=a \omega / 2$, and undergoes instantaneous elastic collision with particle 2 , at time $t_0$. Finally, particles $1$ and $2$ acquire a center of mass speed $v_{ cm }$ and oscillate with amplitude $b$ and the same angular frequency. . . . .

($1$) If the collision occurs at time $t_0=0$, the value of $v_{ cm } /(a \omega)$ will be

($2$) If the collision occurs at time $t_0=\pi /(2 \omega)$, then the value of $4 b^2 / a^2$ will be

Give the answer or quetion ($1$) and ($2$)

A mass $M$ moving with a certain speed $V$ collides elastically with another stationary mass $m$. After the collision, the masses $M$ and $m$ move with speeds $V^{\prime}$ and $v$, respectively. All motion is in one dimension. Then,

Write the principle of conservation of mechanical energy for conservative force.